Re: Comments on Special functions, Quaternions, Octonions

From: Hubert HOLIN (Hubert.Holin_at_[hidden])
Date: 2001-05-29 05:24:47

Somewhere in the E.U., le 29/05/2001

        Bonjour

--- In boost_at_y..., Peter Schmitteckert (boost) <boost_at_s...> wrote:
> Salut,
> 
> > > for x=sqrt( sqrt(epsilon) ), the x*x will be of the order of epsilon,
> > > due to the small prefactor of the next term, you could extend my idea
> > > even to pow( eps,  1/6 ),  when x^6 = epsilon.
> >
> >                OK. I'll add one or more zones where I compute the result via
> > Taylor expansion That really seems necessary. I'll see if the
> > continuity matching zones are really necessary too, however.
> 
> I'm not quite sure if I understand you correctly, what do you exactly
> mean with 'continuity zones' (the plural). One for each function, or several 
> for one function ?

                The terminology may not be the correct one, but what I meant 
was the following:

                If I create a piecewise approximation to a function, say with 
one definition on [0;1] and another on ]1;2], then there is a risk of 
discontinuity at 1, and that's bad if what I intended to model was in 
fact continuous. So instead I find one approwimation on [0;3/2], 
another on [1/2;2], and interpolate between the two approximations on 
[1/2;3/2] (which I called the "continuity" or "matching" zone). This 
generalises readilly.

                We get better (order-0) behaviour, at the cost of more 
computation (especially in the matching zone).

> > > I was only confused, that's all. SInce you introduced the sinc_a family,
> > > I was always searching for one a != pi.
> >
> >                Slated for later, provided this library doe make the cut.
> 
> What is the advatage of sinc_a(x) ? To address rounding problems for very 
> large x ? Then it would be more important to improve sin(x) for very large 
> arguments.

                The main interest of this particular member of the family is 
that it requires fewer computations, and that, in itself it is a 
special function as it is far more common than its brethren.

                That being said, improvements to sin and other common 
functions are indeed needed in legion. Apart from accuracy for large 
values, and the fact that it is not templated (I am still not sure 
::std::sin always picks up the best underlyingimplementation for its 
signature), there are specialized, yet immensely common, occurences 
such as computing a great many values for extremely close values of the 
argument (to draw a circle for instance) that are not taken into 
account (I do have a good solution to this one, though).

                We are just standing at the foot of the numerics pyramid. 
There's work, literraly, for years to come.

> >
> >        Amicalement
> 
> Someone writing sources for octonions had to be a french person.

                ;-)

> Amicalement
> Peter Schmitteckert

        Amicalement

                        Hubert Holin
                        Hubert.Holin_at_[hidden]